Braid Floer Homology on Surfaces

Abstract

The Arnold Conjecture gives the existence of 1-periodic solutions of a nondegenerate Hamiltonian system on a compact symplectic manifold. This Conjecture is proved for aspherical manifolds with the use of Floer homology. After this proof, we continue with developing braid Floer homology on surfaces. We look at free and skeleton braids to define braid Floer homology. The skeleton braids are known solutions for the Hamiltonian. The free braids are unknown solutions. Braid Floer homology tells us when the skeleton forces new solutions. For a special class of skeleton and free braids on the torus, I define braid Floer homology completely.